Signatures of real-space geometry, topology, and metric tensor in quantum transport in periodically corrugated spaces

Abstract

The motion of a quantum particle constrained to a two-dimensional non-compact Riemannian manifold with non-trivial metric can be described by a flat-space Schroedinger-type equation at the cost of introducing local mass and metric and geometry-induced effective potential with no classical counterpart. For a metric tensor periodically modulated along one dimension, the formation of bands is demonstrated and transport-related quantities are derived. Using S-matrix approach, the quantum conductance along the manifold is calculated and contrasted with conventional quantum transport methods in flat spaces. The topology, e.g. whether the manifold is simply connected, compact or non-compact shows up in global, non-local properties such as the Aharonov-Bohm phase. The results vividly demonstrate emergent phenomena due to the interplay of reduced-dimensionality, particles quantum nature, geometry, and topology.

Figures (6)

• Figure 1: Schematics of a two-dimensional Riemannian manifold $(\mathcal{M}, g)$ (marked blue), embedded in higher dimensional space (brown area) and corrugated to a crystal in one dimension with a period $\Lambda^{1}$. The particle moves freely along $(\mathcal{M}, g)$.
• Figure 2: Spatial course of the elementary dent described by \ref{['eq:dent_parametrix']} as well as the geometry-induced effective mass and potential. The setup is characterized by the parameters $m = 0.067\,m_{e}$, $\Lambda^{1}=40\ \mathrm{nm}$, $w=21.5$, and $q_{0}^{1}=20\ \mathrm{nm}$, while $f_{0}$ increases across the rows. The leftmost row refers to flat space which is the reference system.
• Figure 3: Band structure, group velocity and spectral mass as function of $k^1$ for the parameters given in fig. \ref{['fig:Setting']}, with $f_{0}$ increasing across the rows. Different bands are color coded. Band gaps are marked with gray shaded areas. Leftmost row refers to free electron gas in a flat space as reference system.
• Figure 4: Density of states for the corrugation crystals with the same parameters as in fig. \ref{['fig:Setting']}, with $f_{0}$ increasing across the rows. Only one direction is incorporated. Different bands are indicated by color. Spectral regions with gray shade do no host any states and correspond to the band gaps. Leftmost row is for free electron gas.
• Figure 5: Transmittance of a quantum particle traversing a $N$ fold corrugated space. The basis of the repeating structure is formed by corrugations with parameters $m = 0.067\,m_{e}$, $\Lambda^{1} = 40\,\mathrm{nm}$, $q_{0}^{1} = 20\,\mathrm{nm}$, $f_{0} = 8\,\mathrm{nm}$, as well as $w = 21.5$ (dent type A) or $w = 10.0$ (dent type B). (a) Transmittance for a simple lattice structure with basis A. (b) Transmittance for a sub-lattice structure with basis AB.